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Full text of "Mathematical And Physical Papers - Iii"

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63. We have next to investigate the correction for the wire. The effect of the inertia of the air set in motion by the wire was altogether neglected by Bessel, and indeed it would have been quite insensible had the parts of the correction for inertia due to the wire and to the sphere, respectively, been to each other in nearly the same ratio as the parts of the correction for buoyancy. Baily, however, was led to conclude" from his experiments that the effect of the wire was probably not altogether insignificant, and the theory of this paper leads, as we have seen, to the result that the factor n is very large in the case of a very fine wire.
The ivory sphere in Bessel's experiments was swung with a finer wire than the brass sphere. It was for this reason that I did not from the first suppose &/ = &, and &2' = /c2. Let A&, A/^ &c. be the corrections due to the wire. The values of A&ir A&a, A A/, A&8', may be got from the formula (151), in which it is to be remembered that \ denotes the length of the isochronous simple pendulum, not, as in Bessel's notation, the length of the seconds' pendulum. It is stated by Bessel (p. 131), that the wire used with the brass sphere weighed 10*95 Prussian grains in the case of the long pendulum, and 3'58 grains in the case of the short. This gives 7*37 grains for the weight of one toise or 72 French inches. The weight of one toise of the wire employed with the ivory sphere was 6*28 2*04 or 4*24 grains (p. 141). The specific gravity of the wire was 7*6 (p. 40), and the weight of a cubic line (French) of water is about 0'18S5 grain. From these data it results that the radii of the wires were 0*003867 and 0 002933 inch English. The formula (147) gives m, whence L is known from (152). The lengths of the isochronous simple pendulums were about 39'20 inches for the short pendulum, and 116*94 for the long. On substituting the numerical values we get from (151), since ^=1^-1 and &2 = n2~l,
A*, = 0-0107,    AAa = 0-0286,    Ajfc/ = 0'0090,    AA2' = 0'0244.
The specific gravities of the two spheres were about 8190 and 1*794, whence we get from (159) A&=0'0308, or 0*031 nearly.
The value of k deduced by Bessel from his experiments was 0'9459 or 0'946 nearly, which in a subsequent paper he increased to 0'956. In this paper he contemplates the possibility of its being different in the cases of the long and of the short pendulum,